Gerard Freixas i Montplet S eminaire Darboux January...

$Page 1: Gerard Freixas i Montplet S eminaire Darboux January 2020semparis.lpthe.jussieu.fr/contrib/attachments/attach_13301_1.pdf · Gerard Freixas i Montplet C.N.R.S. { Institut de Math$
CY manifolds Mirror families BCOV conjecture The BCOV invariant CY hypersurfaces in PnC

On genus one mirror symmetry

Gerard Freixas i MontpletC.N.R.S. – Institut de Mathematiques de Jussieu - Paris Rive Gauche

Based on joint work with D. Eriksson and C. Mourougane

Seminaire DarbouxJanuary 2020

1 / 41


Calabi–Yau manifolds

2 / 41


Let X be a connected compact Kahler manifold of dimension n.We say that X is a Calabi–Yau (CY) manifold if:

I X has a nowhere vanishing holomorphic differential form, i.e.the canonical bundle is trivial:

KX = ∧nΩX ' OX .

I for 0 < p < n, there are no non-trivial global holomorphicp-forms:

H0(X ,ΩpX ) = 0, or equivalently Hp(X ,OX ) = 0.

A Calabi–Yau manifold is automatically algebraic and projective.

In any given Kahler cohomology class ν ∈ H1,1R (X ), there exists a

unique Ricci flat Kahler metric.

3 / 41


The infinitesimal deformations of the complex structure of X areparametrized by H1(X ,TX ). This has dimension hn−1,1:

H1(X ,TX ) ' H1(X ,TX ⊗ KX ) ' H1(X ,Ωn−1X ).

The deformations of X are unobstructed. There is an openneighborhood of 0, Def(X ) → H1(X ,TX ) and a local universaldeformation of X over Def(X ):

X −→ Def(X ), X0 ' X .

If we fix a polarization L, there is a quasi-projective coarse modulispace M of (X , L). In a neighborhood of the point correspondingto (X , L), M is a quotient of Def(X ) by a finite group.

4 / 41


Let X → S be a morphism of complex manifolds, with Calabi–Yaufibers.

For every s ∈ S , after possibly shrinking S around s, there is aCartesian diagram

X //

X

S // Def(Xs).

We say that the family X → S is maximal if the morphismS → Def(Xs) is a local biholomorphism.

5 / 41


Mirror families

6 / 41


Let X be a CY manifold of dimension n.

Mirror symmetry predicts the existence of a mirror family of CYn-folds ϕ : X∨ → D×:

I D× = (D×)d is a punctured multi-disc, d = hn−1,1(X∨q ).

I ϕ : X∨ → D× is maximal.

I the monodromy T on Rnϕ∗C is maximal unipotent (MUM):if d = 1, this means

(T − 1)n 6= 0 and (T − 1)n+1 = 0.

I mirror Hodge numbers: hp,q(X ) = hn−p,q(X∨q ).

Informally:

ϕ : X∨ → D× ! cusp in a moduli space of CY varieties.

7 / 41


There should also exist a bihomolorphic map, called mirror map,

τ : D× −→ HX := H1,1R (X )/H1,1

Z (X ) + iKX ,

where KX is the Kahler cone of X , which relates the variation ofcomplex structures of X∨ → D× and the Kahler “moduli” of X.

When d = 1, τ can be identified with a multi-valued function ofthe form

τ(q) =1

2πi

∫γ1η∫

γ0η∈ H,

where η is a local holomorphic basis of ϕ∗KX∨/D× and γ0, γ1 arewell-chosen homology n-cycles (Morrison):

I Tγ0 = γ0.

I Tγ1 = γ1 + γ0.

8 / 41


Mirror symmetry (rough version)

Variation of Hodge struc-tures on Rnϕ∗C close to0.

! Curve counting invari-ants in X .

For instance, the mirror symmetry conjecture at genus 0 predictsthat the Yukawa coupling of the mirror family has a holomorphicseries expansion in τ ∈ HX , whose coefficients are given by genuszero Gromov–Witten invariants of X .

In this talk we discuss a mirror symmetry conjecture at genus one,where the role of the Yukawa coupling is played by a functorial liftof the Grothendieck–Riemann–Roch theorem.

9 / 41


Functorial BCOV conjecture

10 / 41


The Grothendieck–Riemann–Roch theorem

Let X be a compact complex manifold.

Let E be a holomorphic vector bundle on X .

Recall the Hirzebruch–Riemann–Roch theorem (HRR):∑q

(−1)q dimHq(X ,E ) =

∫X

ch(E ) td(TX ).

The cohomology groups Hq(X ,E ) can be realized as Dolbeaultcohomology:

Hq

∂(X ,E ) =

ker(∂ : A0,q(E )→ A0,q+1(E )

)Im(∂ : A0,q−1(E )→ A0,q(E )

) .11 / 41


The Grothendieck–Riemann–Roch theorem (GRR) is a variant ofHRR in an algebraic and relative setting.

Let f : X → S be a projective submersion of complex connectedalgebraic manifolds.

Let E be a vector bundle on X .

Theorem (Grothendieck–Riemann–Roch)

The following equality holds in CH•(S)Q:

ch(Rf∗E ) = f∗(ch(E ) td(TX/S)

).

In particular, for the determinant of cohomology:

c1(detRf∗E ) = f∗(ch(E ) td(TX/S)

)(1)in CH1(S)Q ' Pic(S)Q.

12 / 41


In the statement of GRR, Rf∗E stands for the direct image of E inthe derived sense.

When s 7→ dimHq(Xs ,E|Xs ) is constant for all q, the cohomologyspaces Hq(Xs ,E|Xs ) organize into holomorphic vector bundlesRqf∗E . Then

ch(Rf∗E ) =∑q

(−1)q ch(Rqf∗E )

anddetRf∗E =

⊗q

(∧maxRqf∗E )(−1)q.

For instance, this is the case of Hodge bundles, i.e. when we takeE = Ωp

X/S .

13 / 41


Assume now that the fibers Xs are Calabi–Yau (CY).

Define the virtual vector bundle

DΩ•X/S =⊕p

(−1)ppΩpX/S .

HencedetRf∗DΩ•X/S =

⊗(detRqf∗Ω

pX/S)(−1)

p+qp

is a weird combination of determinants of Hodge bundles.

For this datum, GRR simplifies to:

c1(detRf∗DΩ•X/S) =χ

12c1(f∗KX/S) in CH1(S)Q ' Pic(S)Q,

with χ = χ(Xs) the topological Euler characteristic of the fibers.

14 / 41


Definition (BCOV line bundle)

The BCOV line bundle on S is defined by

λBCOV (f ) = detRf∗DΩ•X/S

=⊗p

(detRqf∗ΩpX/S)(−1)

p+qp.

It commutes with arbitrary base change.

Corollary (of GRR)

There exists an isomorphism of Q-line bundles on S

λBCOV (f )⊗12∼−→ (f∗KX/S)⊗χ.

15 / 41


But: there are as many as H0(S ,O×S ) such isomorphisms.

Theorem (Eriksson, Franke)

There exists a canonical isomorphism of Q-line bundles

GRR: λBCOV (f )⊗12∼→ (f∗KX/S)⊗χ,

commuting with arbitrary base change. If f : X → S is definedover Q, GRR is defined over Q as well.

The arithmetic Riemann–Roch theorem of Gillet–Soule provides aweak variant, enough for most purposes:

I natural isomorphism up to a constant of norm one.

I isometry for auxiliary hermitian structures (Quillen metric).

I over Q, the constant is necessarily ±1.

I compatible with Eriksson–Franke.

16 / 41


Mirror symmetry at genus one

Let X be a Calabi–Yau manifold, and ϕ : X∨ → D× its conjecturalmirror family.

Definition

Define the formal generating series of genus one Gromov–Witteninvariants of X by

F1(τ) = − 1

24

∫Xcn−1(X ) ∩ 2πiτ +

∑β

GW1(X , β)e2πi〈β,τ〉,

where

I τ ∈ HX .

I β ∈ H2(X ,Z) runs over curve classes.

I GW1(X , β) = genus one Gromov–Witten invariant of class β.

17 / 41


Conjecture (Optimistic functorial BCOV)

Let X and ϕ : X∨ → D× be mirrors as above.

Assume that ϕ can be extended over an algebraic base.

Let τ : D× → HX be the mirror map.

Then:

I there exist canonical trivializations of λBCOV (ϕ) andϕ∗KX∨/D× .

I in these trivializations, GRR can be identified to aholomorphic function in q ∈ D× of the form

GRR(q) = exp(

(−1)nF1(τ(q)))24

.

18 / 41


The conjecture is optimistic in that:

I we might need to impose further conditions on the Hodgenumbers of X (e.g. hp,q = 0 outside the central cross).

I the usual MUM condition might not be sufficient.

I it could be that the predicted formula for GRR only holds upto a constant.

An alternative would be a conjecture for d log GRR. This stillcaptures the genus one enumerative invariants.

19 / 41


The BCOV invariant

20 / 41


Construction of τBCOV

Let f : X → S be a family of Calabi–Yau manifolds as before.

Recall the canonical GRR isomorphism of Q-line bundles

GRR: λBCOV (f )⊗12∼−→ (f∗KX/S)⊗χ.

To attack the functorial BCOV conjecture we need methods ofexplicitly computing GRR.

Idea:

I introduce natural hermitian metrics on λBCOV (f ) and f∗KX/S

(Hodge theory).

I compute the norm of GRR with respect to these metrics.

I how? Arithmetic Riemann–Roch of Gillet–Soule.

21 / 41


Hermitian metrics via Hodge theory

We endow the line bundle f∗KX/S with the L2 (or Hodge) metric:

hL2,s(α, β) =in

2

(2π)n

∫Xs

α ∧ β,

for α, β are local sections of f∗KX/S .

The normalization (2π)n is standard in Arakelov geometry.

22 / 41


The line bundle λBCOV (f ) has a canonical metric:

I by the Hodge decomposition, there is a C∞ isomorphism

λBCOV (f )⊗2 ⊗ λBCOV (f )⊗2 ∼−→

⊗k

(detRk f∗C)(−1)k2k .

I using the lattice of integral cohomology:⊗k

(detRk f∗C)(−1)k2k =

⊗k

(detRk f∗Z)(−1)k2knt ⊗ C

' C.

The isomorphism is canonical, since (detRk f∗Z)⊗2nt ' Zcanonically.

23 / 41


I All in all, we have a C∞ isomorphism

λBCOV (f )⊗2 ⊗ λBCOV (f )⊗2 ∼−→ C,

which actually defines a smooth hermitian metric onλBCOV (f )⊗2, and hence on λBCOV (f ).

Definition (L2-BCOV metric)

The above canonical hermitian metric on λBCOV (f ) is called theL2-BCOV metric, and denoted hL2,BCOV .

24 / 41


Now the BCOV invariant of the family f : X → S is defined as thenorm of GRR with respect to hL2 and hL2,BCOV :

Definition (BCOV invariant)

We define the BCOV invariant of the family f : X → S as theC∞(S) function

s 7→ τBCOV (Xs) :=‖GRR(θ)‖2L2,s‖θ‖2

L2,BCOV ,s

,

where θ is any local trivialization of λBCOV (f ).

25 / 41


Relation to holomorphic analytic torsion

Assume for simplicity that X has a Kahler form ω, whoserestriction to fibers is Ricci flat.

With respect to ω, we form the ∂-Laplacian ∆p,q

∂,son Ap,q(Xs).

Theorem (Arithmetic Riemann–Roch + ε)

There exists a constant C such that

τBCOV (Xs) = C∏p,q

(det ∆p,q

∂,s)(−1)

p+qpq,

where det ∆p,q

∂,sis the ζ-regularized determinant of ∆p,q

∂,s.

26 / 41


I The arithmetic Riemann–Roch relationship

‖GRR ‖2 = C∏p,q

(det ∆p,q

∂,s)(−1)

p+qpq

is a C∞, spectral evaluation of GRR.

I The original BCOV conjecture was formulated in terms of thefunction

F1(s) :=1

2log∏p,q

(det ∆p,q

∂,s)(−1)

p+qpq.

I Determining the singularities of τBCOV for degenerations ofCY’s is central in our approach to the conjecture. This relieson the spectral interpretation.

27 / 41


The singularities of τBCOV

Let f : Z → D be a germ of a degeneration of CY manifolds:

I f is a projective morphism of complex manifolds, and it canbe extended to a morphism of algebraic varieties.

I f is a submersion over the punctured disc D×, with CY fibers.

Problem: describe the behaviour of t 7→ log τBCOV (Zt) as t → 0.

28 / 41


Here is an attempt of picture of a normal crossings degeneration:

29 / 41


Theorem (CDG 2018)

The following limit exists:

κf := limt→0

log τBCOV (Zt)

log |t|2∈ Q.

We expect that the coefficient κf encodes interesting topologicalinvariants.

We can compute κf when we have a good control on thesingularities of the degeneration.

30 / 41


Theorem (CDG 2018–2019)

I If f acquires at most ordinary quadratic singularities, i.e.modelled on z20 + . . .+ z2n = t, then:

κf =

n+124 # sing(Z0) if n is even,

−n−224 # sing(Z0) if n is odd.

I If KZ = OZ and f is semi-stable, with central fiberZ0 =

∑k Dk a reduced NCD, then

κf =∑k≥1

(−1)kk(k − 1)

24χ(D(k)),

where D(k) = t|J|=kDJ and DJ = Dj1 ∩ . . . ∩ Djk .

31 / 41


The BCOV conjecture for

hypersurfaces in PnC

32 / 41


In the remaining of the talk, we elaborate on the following:

Theorem (CDG 2019)

The functorial BCOV conjecture holds, up to a constant, for CYhypersurfaces in Pn

C and their mirror family.

The 3-dimensional case was obtained by Fang–Lu–Yoshikawa.

First instance of higher dimensional mirror symmetry of BCOVtype at genus one.

It builds on our work on the singularities of τBCOV fordegenerations of CY’s, and some refinements of Schmid’sasymptotics of Hodge metrics.

33 / 41


The mirror family

Let Xn+1 be a general degree n + 1 hypersurface in PnC.

A concrete mirror family f : Z → D× (over a small punctured disc)is constructed in three steps:

I for q ∈ D×, define a CY hypersurface Xq ⊆ PnC by

qn∑

j=0

xn+1j − (n + 1)

n∏j=0

xj = 0.

I kill the symmetries: Yq = Xq/G , where

G = (ξ0, . . . , ξn) ∈ µ×(n+1)n+1 |

∏j

ξj = 1/diagonal.

I Yq is a mildly singular CY variety, but it admits a natural CYdesingularisation Zq → Yq. The construction can be done infamilies, and yields f : Z → D×.

34 / 41


The mirror family can be extended to a morphism of projectivemanifolds Z → P1

C, with some singular fibers:

I the fibers at q ∈ µn+1 have ordinary quadratic singularities(ODP points); each such fiber has a unique singular point.

I the fiber at q = 0 (MUM point), whose geometry we don’tcontrol.

For the BCOV conjecture, we need to canonically trivialize theHodge bundles Rqf∗Ω

pZ/D× .

Relevant Hodge bundles: (Rqf∗ΩpZ/D×)prim with p + q = n − 1.

The relevant Hodge bundles have rank one.

35 / 41


Let F •∞ be the limiting Hodge filtration and W• the monodromyweight filtration, on Hn−1

lim = Schmid’s LMS on (Rn−1f∗C)prim.

For all k = 0, . . . , n − 1,

GrW2k Hn−1lim = GrkF∞ GrW2k H

n−1lim ' GrkF∞ Hn−1

lim

is 1-dimensional. All the other possible graded pieces are trivial.

The flag W• hence looks like

W0 = W1 ⊆W2︸︷︷︸dim 1

= W3 ⊆ . . . ⊆W2n−4 = W2n−3 ⊆W2n−2︸︷︷︸dim 1

= Hn−1lim .

The monodromy weight filtration on (Hn−1)lim ' Hn−1lim has the

same structure.

36 / 41


Theorem

Fix a basis γ• = γkk of (Hn−1)lim adapted to the weightfiltration: γk ∈W ′

2k \W ′2k−2.

Then there exists a unique holomorphic trivialization ϑ• of(Rn−1f∗Ω

•Z/D×)prim, adapted to the Hodge filtration, with

∫γj

ϑk =

0 if j < k

1 if j = k.

Here, the convention is ϑk ∈ Fn−1−k(Rn−1f∗Ω•Z/D×)prim.

37 / 41


Notice that, up to constants, ϑ• only depends on the weightfiltration W• on Hn−1

lim .

Definition

We denote by ηk the trivializing section of (Rk f∗Ωn−1−kZ/D× )prim

obtained by projecting ϑk .

By the theorem and the observation above, the ηk depend only onthe limiting mixed Hodge structure, up to constants.

One can actually show that the ηk extend to trivializations of theDeligne extensions of the Hodge bundles.

38 / 41


Let ‖ · ‖L2 be the L2 norm on (Rk f∗Ωn−1−kZ/D× )prim.

Let q 7→ τ(q) be the mirror map and F1(τ) the generating series ofgenus one Gromov–Witten invariants of Xn+1.

Theorem (CDG 2019)

The BCOV invariant of the mirror family f : Z → D× of Xn+1 hasthe form

τBCOV (Zq) = C∣∣exp

((−1)n−1F1(τ(q))

)∣∣4 ‖Θ‖2,where

‖Θ‖ :=

(‖η0‖χ(Xn+1)/12

L2∏n−1k=0 ‖ηk‖

(n−1−k)L2

)(−1)n−1

and C is some constant.

39 / 41


Some key points of the proof:

I extension of the mirror family to Z → P1C, with controlled

singularities except for q = 0 (MUM point).

I one-dimensional parameter space and complete knowledge ofits meromorphic functions.

I relation of τBCOV to Grothendieck–Riemann–Roch (arithmeticRiemann–Roch in a non-arithmetic setting...).

I behaviour of τBCOV for degenerations with ordinary quadraticsingularities (ODP points).

I good understanding of the sections of the relevant Hodgebundles (explicit constructions, known divisors).

I Zinger’s theorem: computation of F1 in terms ofhypergeometric functions.

40 / 41


From Schmid’s asymptotics for L2 metrics, one derives:

Corollary

As τ → i∞, there is an asymptotic expansion

1

2∂τ log τBCOV (Zq(τ)) = (−1)n−1∂τF1(τ)︸︷︷︸

holomorphic part

+ρ∞Im τ

(1 + O

(1

Im τ

))︸︷︷︸

real analytic in (Im τ)−1

,

where ρ∞ is explicit and depends only on Hn−1lim .

Taking ∂ log removes all the indeterminacies, and produces acanonical expression.

41 / 41

Date post:	19-Oct-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Gerard Freixas i Montplet S eminaire Darboux January...

Documents